Lorentzian band

Regimes I-III are each characterized by peaks immersed in broad-band noise. In regime IV the peaks are replaced by a Lorentzian band noise around [kaneko89]. 

Fig. 13 Isotopic line splitting of the V3 stretching vibration in single crystalline (see also Fig. 12(a)), after [108, 109], The origin of each absorption band is indicated by an isotopomer present in crystals of natural composition. While the absorption could be fitted by a Lorentzian band profile, the remaining peaks were dominated by the Gaussian contribution in the Voigt band shapes (solid lines below the spectrum). The sum result of fitting the isotopic absorption bands is inserted in the measured spectrum as a solid line...

Figure 4. The calculated spectrum of the complex after a Lorentzian band convolution. Region I is dominated by bridging-sulfur-to-iron CT transitions, while region II is mostly due to organic-sulfur-to-iron electron transitions. Regions I and II are explained in a MO diagram. The vertical lines correspond to the experimental bands observed in the absorption spectrum of the [Fe2 (J. - S2) P o - CqH4S) ) ] complex, from Reference 1.

Figure 3.17 presents ps-TR spectra of the olehnic C=C Raman band region (a) and the low wavenumber anti-Stokes and Stokes region (b) of Si-rra i-stilbene in chloroform solution obtained at selected time delays upto 100 ps. Inspection of Figure 3.17 (a) shows that the Raman bandwidths narrow and the band positions up-shift for the olehnic C=C stretch Raman band over the hrst 20-30 ps. Similarly, the ratios of the Raman intensity in the anti-Stokes and Stokes Raman bands in the low frequency region also vary noticeably in the hrst 20-30 ps. In order to better understand the time-dependent changes in the Raman band positions and anti-Stokes/Stokes intensity ratios, a least squares htting of Lorentzian band shapes to the spectral bands of interest was performed to determine the Raman band positions for the olehnic... 

The negative sign in equations 54-19 and 54-20 reflect the fact that the maximum second derivative is a negative value, which also agrees with Figure 54-1, and it also tells us that the magnitude of the second derivative decreases inversely as the cube of a (for the Normal band shape) and inversely as the fifth power of a (for the Lorentzian band shape), that is as the bandwidth of the absorbance band increases. This explains why the derivatives of the broad absorbance band decrease with respect to the narrow absorbance band as we see in Figure 54-1, and more so as the derivative order increases. 

In our previous chapter we derived the expressions for the first and second derivatives of both the Normal and Lorentzian band shapes [1]. For the following discussion, however, we will address only the Normal case, as we will see, the Lorentzian case will parallel it closely. 

Figure 2. Powder emission spectrum of tr-[Rh py ClJ Cl and a lorentzian band analysis yielding three superimposed progressions. Reproduced with permission from Ref. 9. Copyright 1981, VCH-Verlag.

Band profiles were constructed from addition of individual Gaussian-Lorentzian bands using an interactive algorithm (J.Brauner, unpublished results) constructed specially for this purpose. Areas of the component bands were computed and used for calculation of the relative intensities of trans and gaqche conformers. 

It should be noted that due to er(Af)a a the Fourier transform of % (Lorentzian band-shape with the band-width determined by the dephasing constant of different Fourier transform bands are approximately proportional to the FCFs (0aij 0gg 2 (0ai/ 0go) 2- For the single displaced oscillator case, for example, I I2 = S e 2S, where S is the Huang-Rhys factor of this mode. 

Figure 2.22 Calculated and experimental VCD spectra of 18. Spectra of conformations a and b are calculated at the B3LYP/TZ2P level for S-18. Lorentzian band shapes are used (y = 4.0 crrr1). The spectrum of the equilibrium mixture of a and b is obtained using populations calculated from the B3LYP/TZ2P energy difference of a and b. The numbers indicate fundamental vibrational modes. Where fundamentals of a and b are not resolved only the number is shown.

From Eqs. (8) and (10), for a solute band in dilute solution obeying the Beer-Lambert law, the area under a Lorentzian band will be given by... 

In all three methods, the assumptions of a Lorentzian band profile and of a triangular slit function were made. However, since Methods II and III involve measurement over the complete experimental curve, whereas Method I uses only three points of this curve, the latter is the most sensitive to the first assumption. Method II depends upon fairly small corrections related to the band shape and the slit function. Method HI is almost insensitive to the form of the slit function, but is much more strongly dependent upon the assumed band shape. Consequently, for partially overlapping bands, and in general, Methods II and HI are to be preferred, although Method III has speed in its favor. 

Although the selection of an appropriate cut-off frequency value is somewhat arbitrary, various methods of calculating a suitable value have been proposed in the literature. The method of Lam and Isenhour is worth mentioning, not least because of the relative simplicity in calculating/c. The process relies on determining what is termed the equivalent width, EW, of the narrowest peak in the spectrum. For a Lorentzian band the equivalent width in the time domain is given by... 

Figure 5 A Lorentzian band (a), and its first (b), second (c), third (d), and fourth (e) derivatives...

The operator contains a scalar product M, where e is a unit vector in the direction of the coupling electromagnetic vector. Thus, in a well-ordered optically perfect crystal, n must be less than or equal to 2, provided the dispersion on the elements e M i ) and o ly ) is the same. If the transition happens to be permitted for only one orientation of e or if plane-polarized light is used, then n should be 1, thus producing a Lorentzian band contour. 

This spectrum consists of the superposition of two Lorentzian bands... 

By way of introduction let us note that the depolarized spectrum Ivh(co) calculated in Section 7.5 for independent rotors consists of a superposition of Lorentzian bands all centered at zero frequency. In the simplest case of symmetric top rotors the spectrum consists of a single band with a width [q2D + 6<9] which depends only on the translational self-diffusion coefficient D and on the rotational diffusion coefficient 0. This should be compared and contrasted with the depolarized spectrum Ivh(co) of certain pure liquids (e.g., aniline, nitrobenzene, quinoline, hexafluorobenzene) shown schematically in Fig. 12.1.1. The spectrum appears to be split. This entirely novel fea-... 

If Cj(0 is found from Eq. (4.32) and inserted into Eq. (4.25), the absorption spectrum becomes the structureless Lorentzian band envelope ... 

The statistical limit does not require that jj ) be coupled to all the other zero-order states. All that is required for application of the statistical limit is that a sufficiently large number of the ) states are coupled to j ), so that IVR appears to be irreversible. Thus, the experimental observation of a Lorentzian band envelope in an absorption spectrum does not necessarily imply that the initially prepared zero-order state is coupled to all the remaining zero-order states so that a microcanonical ensemble is formed. What is required is that the effective density of states multiplied by the resolution be much larger than one. 

As will be discussed in chapter 6, of fundamental importance in the theory of unimolecular reactions is the concept of a microcanonical ensemble, for which every zero-order state within an energy interval AE is populated with an equal probability. Thus, it is relevant to know the time required for an initially prepared zero-order state j) to relax to a microcanonical ensemble. Because of low resolution and/or a large number of states coupled to i), an experimental absorption spectrum may have a Lorentzian-like band envelope. However, as discussed in the preceding sections, this does not necessarily mean that all zero-order states are coupled to r) within the time scale given by the line width. Thus, it is somewhat unfortunate that the observation of a Lorentzian band envelope is called the statistical limit. In general, one expects a hierarchy of couplings between the zero-order states and it may be exceedingly difficult to identify from an absorption spectrum the time required for IVR to form a micro-canonical ensemble. 

P(n,t) calculated in this manner for the n = 3 overtone of benzene (Lu and Hase, 1988b, 1989) is plotted in Figure 4.13, where it is compared with the quantum result. The classical P(n,t) decays exponentially to zero without the recurrences seen in the quantum calculation. It is these recurrences which give the structure in the absorption spectrum. The spectrum calculated from the classical exponential P(n,t) is a smooth Lorentzian band envelope with fwhm of 85 cm L... 

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